[[Integral domain]]
# Field of fractions

Given an [[integral domain]] $D$, the **field of fractions** $\Frac D$ is the smallest [[field]] into which it can be [[Embedding|embedded]]. #m/def/ring 
Let $D^* = D \setminus \{ 0 \}$.
Then for any $n,m \in D$ and $d,b \in D^*$,
then $\frac{n}{d}, \frac{m}{b} \in \Frac D$ with

1. $$
  \begin{align*}
  \frac{n}{d} = \frac{m}{b} \iff nb = md
  \end{align*}
  $$
2. $$
  \begin{align*}
  \frac{n}{d} + \frac{m}{b} = \frac{nb + md}{db}
  \end{align*}
  $$
3. $$
  \begin{align*}
  \frac{n}{d} \cdot \frac{m}{b} = \frac{nm}{db}
  \end{align*}
  $$

which may be constructed as a quotient of the set $D \times D^*$.
We have the embedding
$$
\begin{align*}
\iota_{D} : D &\hookrightarrow \Frac D \\
n &\mapsto \frac{ns}{s}
\end{align*}
$$
for any $s \in D$.

> [!missing]- Proof of universal property
> #missing/proof

## Universal property

The **field of fractions** of $D$ is a pair consisting of a [[field]] $\Frac D$ and injective [[ring homomorphism]] $\iota : D \hookrightarrow \Frac D$
such that given any field $K$ and injective ring homomorphism $f : D \to K$
there exists a unique ring homomorphism $\bar f : \Frac D \to K$ so that the following diagram commutes

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